Question

If $f\left(x\right)=\left(\frac{p^2-1}{p^2+1}\right)x^3-3x+\log 2$ is a decreasing function of $x$ in $R$, then the set of possible values of $p$(independent of $x$ ) is

• A. $(-1,1)$
• B. $[1,\infty )$
• C. $(-\infty ,-1]$
• D. none of these

Answer 1

The correct option is A $(-1,1)$

For monotonically decreasing function $f^{\prime }\left(x\right)<0$
Therefore differentiating $f\left(x\right)$ with respect to $x$
$\left(\frac{p^2-1}{p^2+1}\right)3x^2-3<0$

$\Rightarrow \left(\frac{p^2-1}{p^2+1}\right)x^2<1$

Since $x^2>0$ for all x

Therefore
$\left(\frac{p^2-1}{p^2+1}\right)<0$
Or
$p^2-1<0$
Or
$p^2<1$
Or
$|p|<1$
Or
$pϵ(-1,1)$.